Remark 8.5 The 2-dimensional case (REF ) shows a stark contrast with the Euclidean space case, since on \(\mathbb {R}^2\)  the Birman-Schwinger principle and the CLR inequality for Schrödinger operators \(\Delta +V\)  do not hold. In particular, recent results of Hoang et al. [1]} show that if \(V\in L_1(\mathbb {R}^2)\setminus 0\)  is such that \(\int V(x)dx\le 0\) , then \(\Delta +V\)  always has at least one negative eigenvalue (see also Simon [2]}).

